ZIMBABWE SCHOOL EXAMINATIONS COUNCIL <br> General Certificate of Education Ordinary Level
**MATHEMATICS**
PAPER 2
4008/2
JUNE 2013 SESSION
2 hours 30 minutes
Additional materials:
Answer paper
Geometrical instruments
Graph paper (3 sheets)
Mathematical tables
Plain paper (1 sheet)
TIME 2 hours 30 minutes
INSTRUCTIONS TO CANDIDATES
Write your name, Centre number and candidate number in the spaces provided on the answer paper/answer booklet.
Answer all questions in Section A and any three questions from Section B.
Write your answers on the separate answer paper provided.
If you use more than one sheet of paper, fasten the sheets together.
**Electronic calculators must not be used.**
All working must be clearly shown. It should be done on the same sheet as the rest of the answer.
Omission of essential working will result in loss of marks.
If the degree of accuracy is not specified in the question and if the answer is not exact, the answer should be given to three significant figures. Answers in degrees should be given to one decimal place.
INFORMATION FOR CANDIDATES
The number of marks is given in brackets [] at the end of each question or part question. Mathematical tables may be used to evaluate explicit numerical expressions.
**This question paper consists of 12 printed pages.**
Copyright: Zimbabwe School Examinations Council, J2013.
**Section A [64 marks]**
Answer all the questions in this section.
Question 1
(a)
Find the exact value of \(20,71-8,2 \times 1,1\).
(b)
Factorise completely
#### (i)
\(12 m-2 n^{2}+6 m n-4 n\),
#### (ii)
\(2 a^{2}-5 a+3\).
(c)
Express as a single fraction in its lowest terms
\[ \frac{x+1}{x-2}-\frac{x+2}{1-x}. \]
Question 2
(a)
Simplify
\[ \frac{1}{3}\left(\begin{array}{cc} -5 & -10 \\ 1 & -1 \end{array}\right)\left(\begin{array}{cc} 1 & 2 \\ 1 & -1 \end{array}\right) \]
(b)
Find the time in which \(\$ 20000\) will earn \(\$ 1600\) simple interest at \(8 \%\) per annum.
(c)
**Take \(\pi\) to be \(\frac{22}{7}\)**
The diagram shows a Compact Disc (CD) for information storage. The useful part is shaded. The inside and outside diameters are 6.6 cm and 13 cm respectively.
Find
#### (i)
the shaded area,
#### (ii)
the percentage area of the disc that is useful.
Question 3
(a)
Solve the equation
\[ \frac{2}{x+2}+\frac{1}{x}=1 . \]
(b)
It is given that \(\mathbf{a}=\binom{3}{3}\) and \(\mathbf{b}=\binom{-1}{2}\).
Find
#### (i)
\(\mathbf{a}+\mathbf{b}\),
#### (ii)
\(\mathbf{a}-3 \mathbf{b}\),
#### (iii)
\(|a-3 b|\).
(c)
Solve the inequality
\(5-3 x \leq 7<-2 x+19\).
Question 4
(a)
In the diagram, ABCD is a cyclic quadrilateral. TC and TDE are tangents to the circle and BC is parallel to \(\mathrm{AT}\). \(\mathrm{ADE}=75^{\circ}\), \(\mathrm{ADC}=58^{\circ}\) and \(\hat{\mathrm{ATD}}=31^{\circ}\).
Find
#### (i)
\(\mathrm{A} \hat{\mathrm{B}} \mathrm{C}\),
#### (ii)
\(\hat{\mathrm{CDT}}\),
#### (iii)
\(\hat{A A D}\),
#### (iv)
ATC ,
#### (v)
\(B \hat{A T}\).
(b)
The surface area, \(A\), of a solid cylinder of height \(h\) and base radius \(r\) is given by the formula \(A=2 \pi r(r+h)\).
#### (i)
Make \(h\) the subject of the formula.
#### (ii)
Find \(h\) when \(A=77 \mathrm{~cm}^{2}, r=2.5 \mathrm{~cm}\) and \(\pi=\frac{22}{7}\).
Question 5
(a)
The dimensions of a rectangle, measured to the nearest centimetre, are 42 cm by 81 cm.
#### (i)
State the least possible width of the rectangle.
#### (ii)
Calculate the least possible perimeter of the rectangle.
(b)
In the diagram, \(\mathrm{A}, \mathrm{B}\) and C are points on level ground. The bearing of B from A is \(016^{\circ}, \mathrm{AB}=4 \mathrm{~km}, \mathrm{BC}=4,5 \mathrm{~km}\) and \(\mathrm{AC}=7 \mathrm{~km}\).
Find
#### (i)
\(\mathrm{BAC}\), giving your answer to the nearest degree,
#### (ii)
the bearing of C from A.
(c)
The volume, \(V\), of a gas at constant temperature is inversely proportional to its pressure \(P\).
#### (i)
Express \(V\) in terms of \(P\) and a constant \(k\).
#### (ii)
Given that \(V=45\) litres when \(P=600\) Newtons per square metre, find \(V\) when \(P=1050\) Newtons per square metre.
Question 6
Answer the whole of this question on a sheet of plain paper.
Use ruler and compasses only and show clearly all construction lines and arcs.
(a)
On a single diagram, construct
#### (i)
quadrilateral ABCD in which \(\mathrm{AD}=\mathrm{BC}=\mathrm{AB}=6 \mathrm{~cm}\), \(\mathrm{DC}=9,5 \mathrm{~cm}\) and \(\mathrm{BCD}=60^{\circ}\),
#### (ii)
the locus of points \(4,5 \mathrm{~cm}\) from A,
#### (iii)
the locus of points equidistant from AD and DC,
#### (iv)
the locus of points equidistant from D and C.
(b)
On your diagram, mark and label the point P which satisfies the loci in (a) (iii) and (iv).
**Section B [36 marks]**
Answer any three questions in this section.
Each question carries 12 marks.
Question 7
A piece of fleece material costs R20 in South Africa and Chido sells each piece at \$5 in Zimbabwe.
(a)
If the price of a piece in South Africa is equivalent to \(\$ 2.50\), calculate the exchange rate between the dollar and the rand.
(b)
Chido bought 20 pieces of fleece in South Africa and incurred \(\$ 30\) in travelling costs. If she sells all 20, calculate her net profit in dollars.
(c)
Each piece of material is enough to make a morning gown. If Chido sews morning gowns and sells each at \(\$ 12\), calculate her net profit from the sale of 20 gowns.
(d)
On another occasion, Chido buys the materials in Zimbabwe at \(\$ 5\) a piece and makes 20 gowns. Calculate the profit she will make without travelling to South Africa.
(e)
If she settles on making gowns, find the difference in the profits realised.
Question 8
The diagram shows the graph of the function \(y=(x+1)^{2}(x-2)\). Use the graph to answer the following questions.
(a)
Write down the roots of the equation \((x+1)^{2}(x-2)=0\).
(b)
Find the coordinates of the points where the gradient of the curve is zero.
(c)
State the range of values of \(x\) for which the function is positive.
(d)
Find the gradient of the curve at the point where \(x=1,5\).
(e)
Use the graph to solve the equation \((x+1)^{2}(x-2)=-2\).
(f)
Find the area bounded by the curve, the \(x\)-axis, the \(y\)-axis and the line
\[ x=1 . \]
Question 9
Answer the whole of this question on a sheet of graph paper.
Triangle \(A\) has vertices at \((2 ; 2),(5 ; 2)\) and \((8 ; 4)\). Using a scale of 2 cm to represent 2 units on both axes, draw the \(x\) and \(y\) axes for \(-8 \leq x \leq 10\) and \(-8 \leq y \leq 8\).
(a)
Draw and label triangle A.
(b)
Triangle \(A\) is mapped onto triangle \(B\) by a translation \(\binom{-9}{2}\). Draw and label triangle B.
(c)
Triangle A is reflected onto triangle C in the line \(\mathrm{y}=-x\). Draw and label triangle C.
(d)
Triangle D with vertices at \((-2 ; 4),(4 ; 4)\) and \((10 ; 8)\) is the image of triangle A under a certain transformation.
#### (i)
Draw and label triangle D.
#### (ii)
Describe completely the single transformation which maps triangle A onto triangle D.
(e)
Draw triangle E, the image of triangle A under a transformation represented by matrix \(\left(\begin{array}{cc}\frac{1}{2} & 0 \\ 0 & 1 \frac{1}{2}\end{array}\right)\).
(f)
Triangle F is the image of triangle A under a clockwise rotation of \(90^{\circ}\) about the point \((2 ;-2)\). Draw and label triangle F.
Question 10
